Image Reconstruction Requirements for Short-Range Inductive Sensors Used in Single-Coil MIT

MIT (magnetic induction tomography) image reconstruction from data acquired with a single, small inductive sensor has unique requirements not found in other imaging modalities. During the course of scanning over a target, measured inductive loss decreases rapidly with distance from the target boundary. Since inductive loss exists even at infinite separation due to losses internal to the sensor, all other measurements made in the vicinity of the target require subtraction of the infinite-separation loss. This is accomplished naturally by treating infinite-separation loss as an unknown. Furthermore, since contributions to inductive loss decline with greater depth into a conductive target, regularization penalties must be decreased with depth. A pair of squared L2 penalty norms are combined to form a 2-term Sobolev norm, including a zero-order penalty that penalizes solution departures from a default solution and a first-order penalty that promotes smoothness. While constraining the solution to be non-negative and bounded from above, the algorithm is used to perform image reconstruction on scan data obtained over a 4.3 cm thick phantom consisting of bone-like features embedded in agarose gel, with the latter having a nominal conductivity of 1.4 S/m.

Towards clinically feasible relaxation-diffusion correlation MRI using MADCO

Multidimensional relaxation-diffusion correlation (REDCO) NMR is an assumption-free method that measures how water is distributed within materials. Although highly informative, REDCO had never been used in clinical MRI applications because of the large amount of data it requires, leading to infeasible scan times. A recently suggested novel experimental design and processing framework, marginal distributions constrained optimization (MADCO), was used here to accelerate and improve the reconstruction of such MRI correlations. MADCO uses the 1D marginal distributions as a priori information, which provide powerful constraints when 2D spectra are reconstructed, while their estimation requires an order of magnitude less data than conventional 2D approaches. In this work we experimentally examined the impact the complexity of the correlation distribution has on the accuracy and robustness of the estimates. MADCO and a conventional method were compared using two T1-D phantoms that differ in the proximity of their peaks, leading to a relatively simple case as opposed to a more challenging one. The phantoms were used to vet the achievable data compression using MADCO under these conditions. MADCO required ~43 and ~30 less data than the conventional approach for the simple and complex spectra, respectively, making it potentially feasible for preclinical and even clinical applications.

Use of marginal distributions constrained optimization (MADCO) for accelerated 2D MRI relaxometry and diffusometry

Measuring multidimensional (e.g., 2D) relaxation spectra in NMR and MRI clinical applications is a holy grail of the porous media and biomedical MR communities. The main bottleneck is the inversion of Fredholm integrals of the first kind, an ill-conditioned problem requiring large amounts of data to stabilize a solution. We suggest a novel experimental design and processing framework to accelerate and improve the reconstruction of such 2D spectra that uses a priori information from the 1D projections of spectra, or marginal distributions. These 1D marginal distributions provide powerful constraints when 2D spectra are reconstructed, and their estimation requires an order of magnitude less data than a conventional 2D approach. This marginal distributions constrained optimization (MADCO) methodology is demonstrated here with a polyvinylpyrrolidone-water phantom that has 3 distinct peaks in the 2D D-T1 space. The stability, sensitivity to experimental parameters, and accuracy of this new approach are compared with conventional methods by serially subsampling the full data set. While the conventional, unconstrained approach performed poorly, the new method had proven to be highly accurate and robust, only requiring a fraction of the data. Additionally, synthetic T1-T2 data are presented to explore the effects of noise on the estimations, and the performance of the proposed method with a smooth and realistic 2D spectrum. The proposed framework is quite general and can also be used with a variety of 2D MRI experiments (D-T2,T1-T2,D-D, etc.), making these potentially feasible for preclinical and even clinical applications for the first time.

Solving 2D Fredholm Integral from Incomplete Measurements Using Compressive Sensing

We present an algorithm to solve the two-dimensional Fredholm integral of the first kind with tensor product structure from a limited number of measurements, with the goal of using this method to speed up nuclear magnetic resonance spectroscopy. This is done by incorporating compressive sensing-type arguments to fill in missing measurements, using a priori knowledge of the structure of the data. In the first step we recover a compressed data matrix from measurements that form a tight frame, and establish that these measurements satisfy the restricted isometry property. Recovery can be done from as few as 10% of the total measurements. In the second and third steps, we solve the zeroth-order regularization minimization problem using the Venkataramanan-Song-Hürlimann algorithm. We demonstrate the performance of this algorithm on simulated data and show that our approach is a realistic approach to speeding up the data acquisition.

Stabilization of the inverse Laplace transform of multiexponential decay through introduction of a second dimension

We propose a new approach to stabilizing the inverse Laplace transform of a multiexponential decay signal, a classically ill-posed problem, in the context of nuclear magnetic resonance relaxometry. The method is based on extension to a second, indirectly detected, dimension, that is, use of the established framework of two-dimensional relaxometry, followed by projection onto the desired axis. Numerical results for signals comprised of discrete T1 and T2 relaxation components and experiments performed on agarose gel phantoms are presented. We find markedly improved accuracy, and stability with respect to noise, as well as insensitivity to regularization in quantifying underlying relaxation components through use of the two-dimensional as compared to the one-dimensional inverse Laplace transform. This improvement is demonstrated separately for two different inversion algorithms, non-negative least squares and non-linear least squares, to indicate the generalizability of this approach. These results may have wide applicability in approaches to the Fredholm integral equation of the first kind.